Dynamical behaviors for generalized pendulum type equations with p-Laplacian

We consider a pendulum type equation with p-Laplacian (ϕp (x′))′ + G′x(t, x) = p(t), where ϕp(u) = ∣u∣p−2u, p > 1, G(t, x) and p(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser’s twist theorem, we find infinitely many invariant tori whenever ∫01p(t)dt = 0, which yields the boundedness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and G′x (t, x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.